Problem: The lifespans of zebras in a particular zoo are normally distributed. The average zebra lives $27.3$ years; the standard deviation is $3.8$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a zebra living less than $34.9$ years.
Explanation: $27.3$ $23.5$ $31.1$ $19.7$ $34.9$ $15.9$ $38.7$ $95\%$ $2.5\%$ $2.5\%$ We know the lifespans are normally distributed with an average lifespan of $27.3$ years. We know the standard deviation is $3.8$ years, so one standard deviation below the mean is $23.5$ years and one standard deviation above the mean is $31.1$ years. Two standard deviations below the mean is $19.7$ years and two standard deviations above the mean is $34.9$ years. Three standard deviations below the mean is $15.9$ years and three standard deviations above the mean is $38.7$ years. We are interested in the probability of a zebra living less than $34.9$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the zebras will have lifespans within 2 standard deviations of the average lifespan. The remaining $5\%$ of the zebras will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({2.5\%})$ will live less than $19.7$ years and the other half $({2.5\%})$ will live longer than $34.9$ years. The probability of a particular zebra living less than $34.9$ years is ${95\%} + {2.5\%}$, or $97.5\%$.